Speaker A
Now that we've talked a little bit about series, I'm going to give you some properties that convergent series display. So much like limits, integrals, and derivatives, convergent series obey these linear properties. In fact, you will have seen these kinds of things if you did manipulations with Riemann sums and computing those. The first thing is, if we have a convergent series, right, and it's important to say that it converges here because otherwise it doesn't really matter. So if we have a convergent series and it converges to some value A, in other words, the sum of the series is this capital A, and C is going to be any real number, okay, then the series given by multiplying every single term of the series by C is basically what I can do is I can factor this constant C out of the infinite series. I really want to emphasize here all of these series are meant to be infinite. The ones you would have dealt with before with the Riemann sums were likely or were all finite series, but here it works for infinite series as well, again, as long as it's convergent. So we can factor the constant out, and in other words, the sum of the series of the new series here is just going to be that constant C times the sum of the old one, right? And again, this is just glorified factoring out a constant from every term. Also, if we have two different series, say this series a_k converges to a and the series of b_k's converges to capital B, then if we make a new series by adding together all the terms or subtracting all the terms, right, so either we sum up a plus b or a minus b, but it's consistent throughout the whole series, then, well, I can split that up, right? In other words, it's kind of like commutative associative. I can add them up however I want. All right, so I can group all the a_k's stuff together and group all the b_k's stuff together and then add or subtract those, right? And so in that way, if I add up all the terms of the sequences or subtract all the terms of the sequences, then, you know, the series, the final thing is still going to be the sum or difference of the actual sums of these series. Okay, lastly, this is quite the important property here, number three. This is, I was mentioning something about the tail of a series. This is what this means here, right? So if m is some positive integer, usually you want to think of it as some really large number, right? So what this says is both of these series either converge or they both diverge. So in particular, like I said, you can ignore basically the first 1 million terms of the sum, toss out all 1 million of those first things, you still have infinitely many things to add up, and the point is it really matters kind of at the end of that series what's going on, right? Because if that limit, if that series doesn't add up to a finite number, it doesn't matter what those first million or billion or quadrillion terms are. Even if the end of that series doesn't add up to a finite number, the whole series is going to be infinite still or diverge somehow, right? So keep that in mind, right? Because even, you know, even if we throw out the first, I don't know, quintillion terms, right, get rid of all of them, there's still infinitely many things to be added. As long as that's finite, if we add in a quintillion extra numbers to that, it doesn't matter because that has to be finite, albeit it might be very large. The point is it must still be finite, right? That's the important thing to keep in mind.
